Borel's paradox

Borel's paradox (sometimes known as the Borel-Kolmogorov paradox) is a paradox of probability theory relating to conditional probability density functions.

A concrete example

A uniform distribution

We are given the joint probability density

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:p_{X,Y}(x,y) =left{egin{matrix} 1, & 0 < y < 1, quad -y < x < 1 - y \ 0, & mbox{otherwise} end{matrix} ight.

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The marginal density of X is calculated to be

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:p_X(x) =left{egin{matrix} 1+x, & -1 < x le 0 \ 1 - x, & 0 < x < 1 \ 0, & mbox{otherwise}end{matrix} ight.

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So the conditional density of Y given X is

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:p_{Y|X}(y|x) =left{egin{matrix} rac{1}{1+x}, & -1 < x le 0, quad -x < y < 1 \ \ rac{1}{1-x}, & 0 < x < 1, quad 0 < y < 1 - x \ \ 0, & mbox{otherwise}end{matrix} ight.

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which is uniform with respect to y.

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Reparametrization

Now, we apply the following transformation:

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:U = rac{X}{Y} + 1 qquad qquad V = Y.

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Using the substitution rule, we obtain

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:p_{U,V}(u,v) =left{egin{matrix} v, & 0 < v < 1, quad 0 < u cdot v < 1 \ 0, & mbox{otherwise} end{matrix} ight.

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The marginal distribution is calculated to be

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:p_U(u) =left{egin{matrix} rac{1}{2}, & 0 < u le 1 \ \ rac {1}{2u^2}, & 1 < u < +infty \ \ 0, & mbox{otherwise}end{matrix} ight.

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So the conditional density of V given U is

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:p_{V|U}(v|u) =left{egin{matrix} 2v, & 0 < u le 1, quad 0 < v < 1 \ 2u^2v, & 1 < u < +infty, quad 0 < v < rac{1}{u} \ 0, & mbox{otherwise}end{matrix} ight.

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which is not uniform with respect to v.

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The unintuitive result

Now we pick a particular condition to demonstrate Borel's paradox. The conditional density of Y given X = 0 is

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:p_{Y|X}(y|x=0) = left{egin{matrix} 1, & 0 < y < 1 \ 0, & mbox{otherwise}end{matrix} ight.

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The equivalent condition in the u-v coordinate system is U = 1, and the conditional density of V given U = 1 is

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:p_{V|U}(v|u=1) = left{egin{matrix} 2v, & 0 < v < 1 \ 0, & mbox{otherwise}end{matrix} ight.

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Paradoxically, V = Y and X = 0 is equivalent to U = 1, but

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:p_{Y|X}(y|x = 0) e p_{V|U}(v|u = 1).

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